Calculating the series expansion of a hyperbolic tangent function to hundreds of terms can be done in seconds.

Mathematica comes with the add‐on package DiscreteMath`RSolve`that allows finding the general terms of series for many functions. After loading this package, and using the package function SeriesTerm, the following term of can be evaluated.

This result can be verified by the following process. The previous expression is not zero for and the corresponding sum has the following value.

Differentiation

Mathematica can evaluate derivatives of the hyperbolic tangent function of an arbitrary positive integer order.

Indefinite integration

Mathematica can calculate a huge set of doable indefinite integrals that contain the hyperbolic tangent function. Here are some examples.

Definite integration

Mathematica can calculate wide classes of definite integrals that contain the hyperbolic tangent function. Here are some examples.

Limit operation

Mathematica can calculate limits that contain the hyperbolic tangent function. Here are some examples.

Solving equations

The next inputs solve two equations that contain the hyperbolic tangent function. Because of the multivalued nature of the inverse hyperbolic tangent function, a message is printed indicating that only some of the possible solutions are returned.

A complete solution of the previous equation can be obtained using the function Reduce.